Athanasius:
The editors of the local newspapers, Boonville Advertiser, Missouri Democrat, and the Central Missouri Republican, were aware that there would have been confusion if there were two Rutherfords in Boonville with the same initials, and would have taken steps to avoid this by using the person's full name or listing his occupation.
That, of course, is just your assumption.
Moreover JFR was not always referred to in the local papers as J. F. Rutherford. Sometimes as Joe Rutherford, attorney Rutherford, Joe F. Rutherford, and once as "Battling Joe" during the 1892 Presidential campaign.
The fact that Joseph was sometimes referred to as things other than J. F. Rutherford has no bearing on anything unless one of those more specific references says anything about membership of the Knights of Pythias.